Distributed Optimization with Inexact Oracle

TL;DR

This paper investigates distributed optimization using inexact first-order oracles, analyzing convergence properties and proposing conditions for exact solutions, supported by numerical experiments.

Contribution

It introduces conditions on inexact oracles that guarantee convergence to the global optimum in distributed settings.

Findings

Distributed subgradient method is suboptimal with certain step sizes.

Conditions on oracle inexactness ensure convergence.

Numerical example verifies the proposed algorithm's efficiency.

Abstract

In this paper, we study the distributed optimization problem using approximate first-order information. We suppose the agent can repeatedly call an inexact first-order oracle of each individual objective function and exchange information with its time-varying neighbors. We revisit the distributed subgradient method in this circumstance and show its suboptimality under square summable but not summable step sizes. We also present several conditions on the inexactness of the local oracles to ensure an exact convergence of the iterative sequences towards the global optimal solution. A numerical example is given to verify the efficiency of our algorithm.